Geodesic
Asymptotics of extremal curves in the ball rolling problem on the plane
Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Mathematical Control Theory and Mechanics (Suzdal, July 3–7, 2009), Tome 42 (2011), pp. 158-165 
A. P. Mashtakov. Asymptotics of extremal curves in the ball rolling problem on the plane. Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Mathematical Control Theory and Mechanics (Suzdal, July 3–7, 2009), Tome 42 (2011), pp. 158-165. http://geodesic.mathdoc.fr/item/CMFD_2011_42_a14/
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     title = {Asymptotics of extremal curves in the ball rolling problem on the plane},
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     volume = {42},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2011_42_a14/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In the present paper, we study an optimal sphere rolling problem on the plane (without slew and slip) with predefined boundary-value conditions. To solve it, we use methods from the optimal control theory. The controlled system for sphere orientation is represented via the rotation quaternion. Asymptotics of extremal paths on a sphere rolling along small-amplitude sine waves is found.

[1] Agrachev A. A., Sachkov Yu. L., Geometricheskaya teoriya upravleniya, Fizmatlit, M., 2005

[2] Arnold V. I., Geometriya kompleksnykh chisel, kvaternionov i spinov, MTsNMO, M., 2002

[3] Lyav A., Matematicheskaya teoriya uprugosti, ONTI, M.-L., 1935

[4] Sachkov Yu. L., “Simmetrii i straty Maksvella v zadache ob optimalnom kachenii sfery po ploskosti”, Matematicheskii sbornik, 201:7 (2010), 99–120 | Zbl

[5] Eiler L., “Metod nakhozhdeniya krivykh linii, obladayuschikh svoistvami maksimuma ili minimuma, ili reshenie izoperimetricheskoi zadachi, vzyatoi v samom shirokom smysle”, Peterburgskoi Akademii Nauk, Prilozhenie I, “Ob uprugikh krivykh”, GTTI, Moskva–Leningrad, 1934, 447–572

[6] Arthur A. M., Walsh G. R., “On the Hammersley's minimum problem for a rolling sphere”, Math. Proc. Cambridge Phil. Soc., 99 (1986), 529–534 | DOI | MR | Zbl

[7] Bicchi A., Prattichizzo D., Sastry S., “Planning motions of rolling surfaces”, IEEE Conf. on Decision and Control, 1995

[8] Jurdjevic V., “The geometry of the plate-ball problem”, Arch. Rat. Mech. Anal., 124 (1986), 305–328 | DOI | MR

[9] Hammersley J. M., “Oxford commemoration ball”, London Math. Soc. Lecture Note Ser., 79, 1983, 112–142 | MR | Zbl

[10] Laumond J. P., Nonholonomic motion planning for mobile robots, LAAS Report 98211, LAAS–CNRS, Toulouse, France, May, 1998

[11] Li Z., Canny J., “Motion of two rigid bodies with rolling constraint”, IEEE Trans. on Robotics and Automation, 6:1 (1983), 62–72

[12] Marigo A., Bicchi A., “Rolling bodies with regular surface: the holonomic case”, Proc. Sympos. Pure Math., 64 (1999), 241–256 | MR | Zbl